# Questions tagged [forcing]

Forcing is a set theoretic method used mainly for proving independence results. For questions about forcing function please use (differential-equations).

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### Examples of first-order claims about the reals that are not preserved under forcing

I am looking for an example of a first-order sentence in the signature of the real numbers, $(+,\times, <, 0,1)$, that is true when translated in the language of set theory in the natural way, but ...
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### Ultrafilters used in Boolean-Valued Forcing

I am beginning to read a paper called "Well-Founded Boolean Ultrapowers as Large Cardinal Embeddings" by Joel David Hamkins and Daniel Evan Seabold. In the first section, they review the ...
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### Why can we not use Cohen forcing to force the continuum to be $\aleph_{\omega}$?

I've gone through the constructions to change add many subsets to different cardinals, and know that Easton's theorem says that the power function can consistently be anything not inconsistent with ...
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### Proof that P-Names allow the generation of all sets derivable from G in M[G]

Kunen says about Forcing : The first step is to define M [G]. Roughly, this will be the set of all sets which can be constructed from G by applying set-theoretic processes definable in M. Each element ...
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### Explanation of last step in proof of Lemma 4.2 in paper "An Axiomatic Approach to Forcing in a General Setting"

The paper "An Axiomatic Approach to Forcing in a General Setting" by R. Freire and P Holy includes the proof of Lemma 4.2. (Note that in the paper a 'generic filter' means a filter in a ...
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### Necessity of Axiom of Choice for unordered pairs of real subsets.

Find a choice function on $\{\{X,Y\} \vert X,Y \in \mathcal{P}(\mathbb{R})\}$ While reading Adrien Douady's book "Algèbre et Théorie Galoisienne", the first chapter focuses on the axiom of ...
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### Linear ordering of reals in Cohen's first model

The presentation of Cohen's first model that I'm most familiar with is to start with a forcing extension by $Add(\omega,\omega)$, consider the group of automorphisms of $Add(\omega,\omega)$ that ...
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### Forcing to add a $\square_\lambda$-sequence is $\mathrm{<}\lambda^+$-strategically closed

For an uncountable cardinal $\lambda,$ a $\square_\lambda$-sequence is a sequence $(C_\alpha: \alpha\in \lim(\lambda^+))$ such that Each $C_\alpha$ is a club in $\alpha$ with order type $\le \lambda.$...
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### Homogeneity of Lévy Collapse

I'm reading the section from Jech's Set Theory regarding the Lévy Collapse $Coll(\aleph_0,<\lambda)$. The following is a lemma towards proving the homogeneity of the Lévy Collapse: Here Jech is ...
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### Questions on Jech's proof of the independence of AC from the ordering principle

In the the book The Axiom of Choice section 5.5, Jech presents a proof of the independence of the axiom of choice from the ordering principle (every set can be linearly ordered). There are some ...
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### Is Forcing only possible in first-order theories?

I'm trying to understand forcing in set theory. From what I understood it necessary that the Löwenheim-Skolem theorem holds to extend models or to force extensions. So, is forcing only possible in ...
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### Foundations of Forcing in Kunen

In Kunen's Set Theory book, forcing is described as a finitistic procedure to get some relative consistency result $Con(ZFC)\Rightarrow Con(T)$, where $T$ is an extension of $ZFC$. Because we can't ...
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### Strange consistency proof based on countability

Let $\sigma(x)$ be a formula (in one free variable $x$) in the language of set theory such that $\mathsf{ZFC} \models \forall x (x \text{ countable } \Rightarrow \sigma(x))$. (E.g., $\sigma(x) =$ &...
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### Borel codes and forcing

Can someone recommend to me some papers or books to learn about coding Borel sets and its absoluteness between transitive models (specially such generated by forcing)? I was reading a paper about ...
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### Is the Forcing Technique an additional independent axiom to ZFC?

Can the Forcing Technique introduced by Cohen be considered to be an axiom or is it a 'technique' with no additional assumptions to ZFC. So does Forcing introduce new objects that are not in V ? The ... Suppose $(\mathbb{P},\leq)$ is a partial order that is (i) separative, i.e. if $x\nleq y$ then $\exists ~z\leq x$ s.t. no $w$ satisfies $w\leq z$ and $w\leq y$ (ii) every strictly descending chain ...
I am reading "Characterization of generic extensions of models of set theory" in https://cmuc.karlin.mff.cuni.cz/pdf/cmuc1703/bukovsky.pdf. Definition 1: For a relation r, let $r''a \iff$ ... 